Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields

TL;DR

This paper applies Iwasawa theory to study Hilbert's tenth problem in $bZ_p$-extensions of imaginary quadratic fields, identifying conditions under which the problem has a negative answer in these infinite towers.

Contribution

It introduces a novel approach using Iwasawa theory to analyze Hilbert's tenth problem for families of number fields in $bZ_p$-extensions of imaginary quadratic fields, identifying specific lines where the problem is unsolvable.

Findings

For primes p=3,11,13,31,37, a positive proportion of imaginary quadratic fields satisfy the criteria.

Identifies a line in $bZ_p$-extensions where Hilbert's tenth problem has a negative answer.

Uses explicit elliptic curves and recent results to establish the criteria.

Abstract

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Kriz--Li and Bhargava et al., we demonstrate that for primes $ p = 3, 11, 13, 31, 37 $, a positive proportion of imaginary quadratic fields meet our criteria.